Simplify the following expression and state the condition under which the simplification is valid. You can assume that $x \neq 0$. $t = \dfrac{3x - 4}{-9} \div \dfrac{6x(3x - 4)}{2x} $
Dividing by an expression is the same as multiplying by its inverse. $t = \dfrac{3x - 4}{-9} \times \dfrac{2x}{6x(3x - 4)} $ When multiplying fractions, we multiply the numerators and the denominators. $t = \dfrac{ (3x - 4) \times 2x } { -9 \times 6x(3x - 4) } $ $ t = \dfrac {2x (3x - 4)} {-9 \times 6x(3x - 4)} $ $ t = \dfrac{2x(3x - 4)}{-54x(3x - 4)} $ We can cancel the $3x - 4$ so long as $3x - 4 \neq 0$ Therefore $x \neq \dfrac{4}{3}$ $t = \dfrac{2x \cancel{(3x - 4})}{-54x \cancel{(3x - 4)}} = -\dfrac{2x}{54x} = -\dfrac{1}{27} $